Graph Saturation in Color
Michael Ferrara
Jaehoon Kim
Elyse Yeager*
yeager2@illinois.edu
UAA Mathematics Colloquium
February 2015
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
1 / 22
Outline
1
Introduction to Graphs
2
Graph Saturation
3
Ramsey Theory
4
Colored Graph Saturation
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
2 / 22
What Do You Know About Graphs?
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
3 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
vertex
|G | : number of vertices in G
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
vertex
|G | : number of vertices in G
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
edge
vertex
|G | : number of vertices in G
kG k : number of edges in G
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Graph Basics
Subgraph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
4 / 22
Common Graphs
Path:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Clique:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Clique:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Clique:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Common Graphs
Path:
Matching:
Clique:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
5 / 22
Outline
1
Introduction to Graphs
2
Graph Saturation
3
Ramsey Theory
4
Colored Graph Saturation
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
6 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
This graph is triangle saturated
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
This graph is triangle saturated
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
This graph is triangle saturated
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Definition
A graph G is H-saturated if:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Definition
A graph G is H-saturated if:
1
H is not a subgraph of G
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Goal:
Avoid a forbidden subgraph, but add as many edges as possible.
Example:
Let the triangle be forbidden.
Definition
A graph G is H-saturated if:
1
H is not a subgraph of G and
2
If we add any edge to G , H is a subgraph of the resulting graph.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
7 / 22
Saturation Number
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
8 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
8 edges
5 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
8 edges
5 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
8 edges
5 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Given the above examples:
sat(6; triangle)
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Saturation Number
7 edges
8 edges
5 edges
Definition: (Erdős-Hajnal-Moon)
The saturation number of a graph H and a number n is the minimum
number of edges in a graph on n vertices that is H saturated.
sat(n; H) := min{kG k : |G | = n and G is H saturated}
Given the above examples:
sat(6; triangle) ≤ 5
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
8 / 22
Outline
1
Introduction to Graphs
2
Graph Saturation
3
Ramsey Theory
4
Colored Graph Saturation
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
9 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Colors: red, blue.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Colors: red, blue.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Colors: red, blue.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Edge Coloring
Definition:
An edge coloring of a graph is an assignment of a color to each edge.
Definition:
A graph is monochromatic if all its edges are assigned the same color.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
10 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
11 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Example:
Color red and blue; forbid red triangle and blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
11 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Example:
Color red and blue; forbid red triangle and blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
11 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Example:
Color red and blue; forbid red triangle and blue triangle.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
11 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Example:
Color red and blue; forbid red triangle and blue triangle.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
bad coloring
Graph Saturation in Color
Feb 2015
11 / 22
Ramsey Problems
Goal:
An edge-coloring with no forbidden monochromatic subgraph.
Example:
Color red and blue; forbid red triangle and blue triangle.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
bad coloring
Graph Saturation in Color
bad coloring
Feb 2015
11 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Example: Any red-blue coloring of the edges of K6 contains a
monochromatic red triangle or a monochromatic blue triangle.
no good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ramsey’s Theorem
Given any collection of forbidden subgraphs, and any sufficiently large
clique, no good coloring exists.
The lowest “sufficiently large” number for a given collection of subgraphs
is called the Ramsey Number.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ramsey’s Theorem
Given any collection of forbidden subgraphs, and any sufficiently large
clique, no good coloring exists.
The lowest “sufficiently large” number for a given collection of subgraphs
is called the Ramsey Number.
R(3, 3) = 6 (two triangles)
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ramsey’s Theorem
Given any collection of forbidden subgraphs, and any sufficiently large
clique, no good coloring exists.
The lowest “sufficiently large” number for a given collection of subgraphs
is called the Ramsey Number.
R(3, 3) = 6 (two triangles)
R(4, 4) = 18
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ramsey’s Theorem
Given any collection of forbidden subgraphs, and any sufficiently large
clique, no good coloring exists.
The lowest “sufficiently large” number for a given collection of subgraphs
is called the Ramsey Number.
R(3, 3) = 6 (two triangles)
R(4, 4) = 18
43 ≤ R(5, 5) ≤ 49
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Ramsey Problems
Note:
Sometimes a good coloring doesn’t exist!
Ramsey’s Theorem
Given any collection of forbidden subgraphs, and any sufficiently large
clique, no good coloring exists.
Erdős-Szekeres: Given N, any sufficiently large collection of points in
general position contains a subset forming the vertices of a convex N-gon
Van der Waerden: Any coloring of the natural numbers contains
arbitrarily long monochromatic arithmetic sequences.
Green-Tao: The sequence of prime numbers contains arbitrarily long
arithmetic progressions.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
12 / 22
Outline
1
Introduction to Graphs
2
Graph Saturation
3
Ramsey Theory
4
Colored Graph Saturation
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
13 / 22
Marrying The Two
Recall:
A graph G is H-saturated if G contains no H subgraph, but adding any
edge to G creates an H subgraph.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
14 / 22
Marrying The Two
Recall:
A graph G is H-saturated if G contains no H subgraph, but adding any
edge to G creates an H subgraph.
Ramsey version of saturation:
Given forbidden graphs H1 , . . . , Hk (in colors 1, . . . , k respectively), we say
a graph G is (H1 , . . . , Hk )-saturated if a good edge-coloring of G exists
(using colors 1, . . . , k), but if we add any edge to G , all colorings are bad.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
14 / 22
Marrying The Two



Example: the graph below is 
Ferrara-Kim-Yeager (UCD, UIUC)
,
Graph Saturation in Color

-saturated.
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
Marrying The Two



Example: the graph below is 
,

-saturated.
First: show a good coloring exists.
Second: show no good coloring exists if we add any edge.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
15 / 22
rsat(n; H1 , . . . , Hk )
Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few
edges as possible.
Definition:
For a number n and forbidden subgraphs H1 , . . . , Hk , we define
rsat(n; H1 , . . . , Hk )
to be the minimum number of edges over all n-vertex graphs that are
(H1 , . . . , Hk )-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
16 / 22
rsat(n; H1 , . . . , Hk )
Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few
edges as possible.
Definition:
For a number n and forbidden subgraphs H1 , . . . , Hk , we define
rsat(n; H1 , . . . , Hk )
to be the minimum number of edges over all n-vertex graphs that are
(H1 , . . . , Hk )-saturated.
Previous Example:

is 
Ferrara-Kim-Yeager (UCD, UIUC)

,
-saturated
Graph Saturation in Color
Feb 2015
16 / 22
rsat(n; H1 , . . . , Hk )
Again, we are interested in (H1 , . . . , Hk )-saturated graphs with as few
edges as possible.
Definition:
For a number n and forbidden subgraphs H1 , . . . , Hk , we define
rsat(n; H1 , . . . , Hk )
to be the minimum number of edges over all n-vertex graphs that are
(H1 , . . . , Hk )-saturated.
Previous Example:

is 
Ferrara-Kim-Yeager (UCD, UIUC)

,

-saturated, so rsat 5;
Graph Saturation in Color

,
 ≤ 6.
Feb 2015
16 / 22
Forbidding Cliques: Ramsey Numbers
Definition:
Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for
forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
17 / 22
Forbidding Cliques: Ramsey Numbers
Definition:
Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for
forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists.
Example: R(3, 3) = 6
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
17 / 22
Forbidding Cliques: Ramsey Numbers
Definition:
Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for
forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists.
Example: R(3, 3) = 6
no good coloring of K6 exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
17 / 22
Forbidding Cliques: Ramsey Numbers
Definition:
Let R = R(c1 , . . . , ct ) be the smallest natural number so that, for
forbidden cliques Kc1 , . . . , Kct , no good coloring of KR exists.
Example: R(3, 3) = 6
no good coloring of K6 exists
Ferrara-Kim-Yeager (UCD, UIUC)
a good coloring of K5 exists
Graph Saturation in Color
Feb 2015
17 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
Ferrara-Kim-Yeager (UCD, UIUC)
n − (R − 2)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
If we add any edge, NO good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
If we add any edge, NO good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
A good coloring exists
If we add any edge, NO good coloring exists
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Forbidding Cliques: Hanson-Toft
Let R = R(c1 , . . . , ct ). The construction below is (Kc1 , . . . , Kct )-saturated.
KR−2
n − (R − 2)
Hanson-Toft
Conjecture: The construction above has the fewest possible edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
18 / 22
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Graph Saturation in Color
Feb 2015
19 / 22
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Graph Saturation in Color
Feb 2015
19 / 22
Hanson-Toft
Hanson-Toft Conjecture
sat(n; Rmin (Kk1 , . . . , Kkt )) =
r −2
2
n
2
n<r
+ (r − 2)(n − r + 2) n ≥ r
Chen, Ferrara, Gould, Magnant, Schmitt; 2011
sat(n; Rmin (K3 , K3 )) =
Ferrara-Kim-Yeager (UCD, UIUC)
n
2
n<6=r
4n − 10 n ≥ 56
Graph Saturation in Color
Feb 2015
19 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Matching Saturation: Mader, 1973
Forbidden:
Graph:
No forbidden subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Adding any edge creates
forbidden subgraph
Graph Saturation in Color
Feb 2015
20 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y
Example: Forbidden matching on 10 edges.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Forbidding Matchings: Ferrara-Kim-Y


Colored Example: Forbidden 
Ferrara-Kim-Yeager (UCD, UIUC)

,
,
Graph Saturation in Color
,

.
Feb 2015
21 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
Ferrara-Kim-Yeager (UCD, UIUC)
,

.
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
Ferrara-Kim-Yeager (UCD, UIUC)
,

.
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
This (uncolored!) subgraph is saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
This (uncolored!) subgraph is saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
This (uncolored!) subgraph is saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Iterated Recoloring: Ferrara-Kim-Yeager
Goal:
Use results from (uncolored) saturation in the Ramsey version.



Example: Forbidden graphs 
,

.
good coloring
make red-heavy
take red subgraph
This (uncolored!) subgraph is saturated.
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
22 / 22
Thanks for Listening!
Ferrara-Kim-Yeager (UCD, UIUC)
Graph Saturation in Color
Feb 2015
23 / 22